There is a simple way of improving the generation of dual codewords of low weight on \({{N}}\) . It consists in partitioning \({{N}}\)  in two sets \({{N}}_1\)  and \({{N}}_2\)  with
\({{N}}_2\)  being a subset of positions of size just a little bit above \(n-k\)  (which is the dimension of the dual code \(\perp \) ), say \(n-k+\ell \)  and then to use the collision method to get dual codewords of weight \(w_2\)  on \({{N}}_2\) . The same method is used in the improvement [1]} of the simple collision decoder [2]} or in a slightly less efficient way in [3]}. 

It just consists in finding codewords in \(\perp \)  which have weight \(w_1\)  on \({{N}}_1\)  and \(w_2\)  on \({{N}}_2\)  instead of simply weight \(w\)  on \({{N}}\) . We have represented in Figure REF  the form of the parity-checks we produce with this method. Note that the weight \(w_1\)  is expected to be half the size \(k-\ell -s\)  of \({{N}}_1\) .
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